How does alcohol affect the brain?

This is a great question, but a very complex one! There are many short-term and long-term effects of alcohol on the brain.
Primarily, alcohol is a central nervous system depressant, which means that it suppresses the regulatory functions of the brain, leading to the classical slurred speech and poor limb coordination often seen amongst people who are drunk. This is due to the effect of alcohol on chemicals called neurotransmitters within the brain, such as GABA, glutamate and dopamine. Alcohol can affect all areas of the brain, including the meso-limbic pathway, the cerebellum, the hippocampus and pre-frontal cortex, leading to changes in coordination, mood, movement and speech.
Extreme examples of long-term alcohol abuse can lead to conditions called Wernicke's Encephalopathy and Korsakoff's Psychosis.